Gov. Sam Brownback and Kansas legislators appear headed toward a stalemate over spending on public schools that could block any significant increase in aid or change in how dollars are distributed.

Brownback has proposed overhauling how the state parcels out more than $3 billion — roughly half its general tax revenues — to its 286 school districts. His plan is designed to make the funding formula simpler, loosen constraints on local boards of education and direct more of the state’s dollars into classrooms.

The governor has argued that an overhaul is necessary to end litigation against the state. An education funding lawsuit is set to go to trial June 4 in Shawnee County District Court.

But many legislators saw rewriting the funding formula as too daunting a task for an annual, 90-day session already crowded with other big issues, and lawmakers now expect to study the governor’s proposal over the summer and fall. Democrats and some Republicans also argue the funding formula works with enough money, and they’re seeking to reverse some cuts made in recent years.

A bill to increase state aid to public schools by $156 million over two years cleared the Senate last month with bipartisan support, but it’s received a cool reception from Brownback’s fellow conservative Republicans who control the House.

Senate Education Committee Chairwoman Jean Schodorf, a Wichita Republican, also acknowledged that legislators would have to reconcile a spending increase with proposals pursued by both Brownback and lawmakers to cut taxes. The Legislature returns April 25 from its annual spring break to wrap up the year’s business.

“All the balls are in the air, and school finance will be one of the last balls to hit,” Schodorf said. “The wild card is what the governor would do with it.”

Brownback pushed legislators last year to cut the state’s general aid to schools to help balance the budget, yet he proposed restoring about $90 million over the next two school years. But he tied the extra money to having dramatic changes in the school funding formula take effect for 2013-14.

Brownback’s plan would scrap a two-decade-old practice of linking some of districts’ spending authority to the number of students at risk of failing or the number who don’t speak English well. It also would eliminate limits on school districts’ power to increase property taxes.

“The Senate’s proposal does not address the fundamental problem facing our state, which is getting school finance out of the courts,” said Brownback spokeswoman Sherriene Jones-Sontag. “More money without reform is not the solution.”

And House Speaker Mike O’Neal, a Hutchinson Republican and Brownback ally, isn’t sold on increasing aid to public schools even if the state overhauls its funding formula. He and some other conservatives contend districts could have avoided cutting programs and laying off teachers and other employees by tapping cash reserves. The reserves have grown: As of July 2011, they exceeded $1.7 billion and were 38 percent higher than five years before.

“They ought to use the money we already appropriated for educating kids before getting more,” O’Neal said.

General aid for schools peaked in the 2008-09 school year at $4,400 per student, having increased dramatically following Kansas Supreme Court rulings in 2005 and 2006 that the state’s wasn’t distributing its dollars fairly or spending enough money to live up to constitutional responsibilities to provide a suitable education for every child. But since then, the figure has dropped to $3,780 per student, with Brownback winning a cut of nearly 6 percent alone last year.

The lawsuit scheduled for trial in June was filed by more than 30 students, their parents and guardians and the Dodge City, Hutchinson, Kansas City and Wichita school districts.

Supporters of the Senate’s plan argue that it represents a good-faith effort to address issues raised by the lawsuit. The measure also allows school districts a slight increase in their property taxes, tied to extra dollars for poorer districts.

“The whole goal in all this is to start restoring the cuts,” said Senate Minority Leader Anthony Hensley, a Topeka Democrat. “The Legislature hasn’t held up its end of the deal.”

House Education Committee Chairman Clay Aurand, a Belleville Republican, said of the push for additional dollars: “I would think there might be a chance to put a little more in.”

But O’Neal remained skeptical, saying questions about the fairness of how dollars are distributed are more important.

“I don’t know that we have an adequacy problem. We may have an equity problem,” he said. “That money that the Senate is spending, it would only exacerbate the problems.”